Symmetry and bifurcation in three-dimensional elasticity, II

Chillingworth, D. R. J., J. E. Marsden and Y. H. Wan

Arch. Ration. Mech. Anal., 83, (1983), 363-395

Abstract:

In part I of this paper (Chillingworth, Marsden and Wan [1982]--hereafter referred to as [1]), we reformulated the traction problem in elastostatics in various forms, gave a classification of loads and gave complete analysis of solutions of the traction problem that are nearly stress-free for loads near loads of type 0 and type 1. This part develops the basic theory as well as giving an analysis of solutions for loads of types 2, 3 and 4. It includes a count of the numbers of solutions and an analysis of their stability and the structural stability of the bifurcation diagrams.
We begin in Section 2 with a derivation of a potential formulation of the problem on SO
(3). The "second order potential" used in [1] can be recovered as a special case. It follows from this that the traction problem always has at least four solutions, at least one of which is neutrally stable. For loads of type 0, we showed in [1] that there are exactly four solutions near SO
(3); for the other types there can be many more ... up to 40. Section 3, 4 and 5 examine types 2, 3 and 4 respectively, in a manner analogous to our treatment of types 0 and 1 in [1]. Loads of type 3 and 4 have some special features already studied in the literature in connection with parallel loads. These special features will be discussed and other connections with the existing literature will be made at appropriate points throughout the paper.